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Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"certificate_hash":"14dce620e8e2fd9a39f71677e181f4d036053ee5f3d65b9ee823423804e8b3b7","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-1e7342ec7a250847","role":"sequence","name":"shift_x_plus_one of Triangle read by rows: T(n,k) is k-th entry of the gamma-vector of the n-dimensional cyclohedron, 0 <= k <= floor(n/2).","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-3d8de1f247a3730f","kind":"fingerprint_related","node_a_id":"n-6b5aa313c13a9a3e","node_b_id":"n-2652387edb474022","fingerprint":"f6-607fceb0be5e82a1","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-607fceb0be5e82a1"},"other_id":"n-6b5aa313c13a9a3e","other":{"id":"n-6b5aa313c13a9a3e","name":"theta_plus_one_squared of shift_x_plus_one of T(n,k) = binomial(n,2*k)*binomial(2*k,k) for 0 <= k <= n, triangle read by rows.","first_index":1,"fingerprint":"f6-607fceb0be5e82a1","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["3","8"]],"card_rows":[["1"],["1"],["3","8"],["7","24"]],"row_sums_preview":["1","1","11","31","169","641","2891","11831"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4b168caf0bc63f8b","steps":["x \u21a6 x+1","(\u03b8+1)\u00b2"],"edges":["e-502ba5da9494e9e3","e-559af0bef77ea39a"],"text":"A089627 \u2192 x \u21a6 x+1 \u2192 (\u03b8+1)\u00b2"},"headline":"A089627 \u2192 x \u21a6 x+1 \u2192 (\u03b8+1)\u00b2","chain":null}}],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:    1;\n  1:    1;\n  2:    3,     8;\n  3:    7,    24;\n  4:   19,    96,    54;\n  5:   51,   320,   270;\n  6:  141,  1080,  1350,   320;\n  7:  393,  3528,  5670,  2240;\n  8: 1107, 11424, 22680, 13440,  1750;\n  9: 3139, 36576, 86184, 67200, 15750;\n","recurrence_sentence":"The row polynomials satisfy (36*n+(60*n^2+33*n)*x+(46*n^3+22*n^2+n)*x^2+(12*n^4+18*n^3-6*n^2-10*n)*x^3+(n^5+2*n^4+n^3)*x^4)*P(n,x) = (180*n-144+(426*n^2+123*n-456)*x+(302*n^3+676*n^2-389*n-520)*x^2+(72*n^4+454*n^3+258*n^2-506*n-264)*x^3+(5*n^5+86*n^4+185*n^3+4*n^2-204*n-72)*x^4+(4*n^5+20*n^4+28*n^3-4*n^2-32*n-16)*x^5)*P(n-1,x) - (54+(216*n+270)*x+(222*n^2+906*n+558)*x^2+(72*n^3+770*n^2+1502*n+626)*x^3+(6*n^4+204*n^3+962*n^2+1252*n+420)*x^4+(14*n^4+180*n^3+518*n^2+536*n+168)*x^5+(8*n^4+48*n^3+104*n^2+96*n+32)*x^6)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((180*n - 144 + (426*n^2 + 123*n - 456)*x + (302*n^3 + 676*n^2 - 389*n - 520)*x^2 + (72*n^4 + 454*n^3 + 258*n^2 - 506*n - 264)*x^3 + (5*n^5 + 86*n^4 + 185*n^3 + 4*n^2 - 204*n - 72)*x^4 + (4*n^5 + 20*n^4 + 28*n^3 - 4*n^2 - 32*n - 16)*x^5)*P[n - 1, x] - (54 + (216*n + 270)*x + (222*n^2 + 906*n + 558)*x^2 + (72*n^3 + 770*n^2 + 1502*n + 626)*x^3 + (6*n^4 + 204*n^3 + 962*n^2 + 1252*n + 420)*x^4 + (14*n^4 + 180*n^3 + 518*n^2 + 536*n + 168)*x^5 + (8*n^4 + 48*n^3 + 104*n^2 + 96*n + 32)*x^6)*D[P[n - 1, x], x])/(36*n + (60*n^2 + 33*n)*x + (46*n^3 + 22*n^2 + n)*x^2 + (12*n^4 + 18*n^3 - 6*n^2 - 10*n)*x^3 + (n^5 + 2*n^4 + n^3)*x^4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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